Central product

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In mathematics, especially in the field of group theory, the central product is way of producing a group from two smaller groups. The central product is similar to the direct product, but in the central product two isomorphic central subgroups of the smaller groups are merged into a single central subgroup of the product. Central products are an important construction and can be used for instance to classify extraspecial groups.

[edit] Definition

A group G is the central product of two of its subgroups G₁ and G₂ if the following hold:

G = G₁ G₂
For all $g_1 \in G_1$ and $g_2 \in G_2, g_1 g_2 = g_2 g_1$
$G_1 \cap G_2$ is contained in the center of G.

A sufficient condition for G to be the central product of G₁ and G₂ is that G is isomorphic to the direct product of G₁ and G₂.

[edit] Examples

Every extraspecial group is a central product of extraspecial groups of order p³.

[edit] References

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